Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

TL;DR

This paper extends criticality theory to the $ ext{A}$-Laplace equation with a potential, establishing fundamental spectral properties, eigenvalue uniqueness, and solutions' behavior at infinity in a broad setting.

Contribution

It introduces a generalized criticality framework for the $ ext{A}$-Laplace operator with potential, including an AAP theorem and eigenvalue characterization.

Findings

Proved an Agmon-Allegretto-Piepenbrink type theorem.

Established the uniqueness and simplicity of the principal eigenvalue.

Analyzed positive solutions of minimal growth and Green functions.

Abstract

In this paper, we study positive solutions of the quasilinear elliptic equation $$Q'_{p,\mathcal{A},V}[u]\triangleq-\mathrm{div}{\mathcal{A}(x,\nabla u)}+V(x)|u|^{p-2}u=0,$$ in a domain $\Omega\subseteq \mathbb{R}^n$, where $n\geq 2$, $1<p<\infty$, the divergence of $\mathcal{A}$ is the well known $\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpel\"{a}inen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q'_{p,\mathcal{A},V}$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q'_{p,\mathcal{A},V}$ in a domain $\omega\Subset\Omega$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q'_{p,\mathcal{A},V}[u]=0$ of minimal growth at infinity in $\Omega$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.